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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Analytic perturbation of the Taylor spectrum


Author: Zbigniew Slodkowski
Journal: Trans. Amer. Math. Soc. 297 (1986), 319-336
MSC: Primary 47A56; Secondary 32A99, 47A10, 47D99
DOI: https://doi.org/10.1090/S0002-9947-1986-0849482-9
MathSciNet review: 849482
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Abstract: Let ${T_1}(z), \ldots ,{T_m}(z)$, $z \in G \subset {{\mathbf {C}}^k}$, be analytic families of bounded operators in a complex Banach space $X$, such that for each $z \in G$ the operators ${T_i}(z)$ and ${T_j}(z)$, $i,j = 1, \ldots ,n$, commute. Main result: If $K(z)$ denotes the Taylor spectrum of the tuple $({T_1}(z), \ldots ,{T_m}(z))$, then the set-valued function $K:G \to {2^{{\mathbf {C}}m}}$ is analytic. Analyticity of such set-valued functions is defined here by a simultaneous local maximum property of $k$-tuples of complex polynomials on the graph of $K$.


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Keywords: Analytic multifunction, local maximum property, joint spectrum, Taylor spectrum, exact complex, analytic perturbation
Article copyright: © Copyright 1986 American Mathematical Society